A trace formula derivation of the functional equation for symmetric square L-functions

Dongsheng Wu; Taiwang Deng

REVIEW 3 major objections 5 minor 19 references

The symmetric-square L-function of a level-one holomorphic or even Maass cusp form is entire and satisfies a s ↦ 1−s functional equation that follows directly from Petersson and Kuznetsov trace formulas via an averaged Voronoi identity.

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2026-08-02 11:21 UTC pith:GWVVFLPC

load-bearing objection The averaged Voronoi identities are the right idea, but Lemma 3.6 is numerically false, so the square-ℓ correction collapses. the 3 major comments →

arxiv 2606.15729 v2 pith:GWVVFLPC submitted 2026-06-14 math.NT

A trace formula derivation of the functional equation for symmetric square L-functions

Taiwang Deng , Dongsheng Wu This is my paper

classification math.NT MSC 11F7211F6611F68

keywords symmetric-square L-functionfunctional equationtrace formulaPetersson formulaKuznetsov formulaVoronoi summationMaass formsMellin inversion

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper claims that for level-one holomorphic cusp forms of even weight and for even Hecke–Maass cusp forms, the symmetric-square L-function extends to an entire function and obeys a functional equation under s ↦ 1−s, and that this symmetry can be derived from Petersson and Kuznetsov trace formulas alone, without using the known lift from GL(2) to GL(3). The proof route builds an averaged Voronoi reciprocity identity: after inserting symmetric-square coefficients on the spectral side, Poisson summation in a quadratic variable produces quadratic Dirichlet series, and the archimedean factors are identified by a Weber–Schafheitlin calculation. Mellin inversion and a spectral-separation argument then recover the functional equation with the expected gamma factors and root number +1. If correct, the paper supplies a concrete template showing how L-functions beyond the standard case can emerge from a trace-formula comparison, and the mechanism applies uniformly to the Maass case including the continuous spectrum.

Core claim

On the paper's own terms, the discovery is the averaged identity I_ℓ(g)=I_ℓ(T_μg) for every test function g and every positive integer ℓ: the weighted sum of symmetric-square coefficients against g is unchanged when the test function is replaced by the archimedean transform T_μ, whose Mellin multiplier is G_μ(s)=G_0(s+2k−1)G_0(s−2k+1)G_0(s). From this identity, and its Maass analogue with an explicit Eisenstein correction, the paper derives entireness and the functional equation Λ(s)=Λ(1−s) for each individual L(s,Sym²f) by choosing the ℓ's to separate the finite-dimensional space of eigenforms. The root number is +1 and the gamma factor matches the classical one; the derivation avoids any a

What carries the argument

The averaged Voronoi identity is the load-bearing object. The archimedean transform T_μ is the central mechanism: it is defined by Mellin inversion with multiplier G_μ(s), chosen so that G_μ(s)G_μ(1−s)=1 and so that, after Poisson summation, the kernel J_{n,ℓ}(z) satisfies the same functional equation as the quadratic Dirichlet series L_{n²−4ℓ}(z). The Weber–Schafheitlin integral supplies the explicit form of J_{n,ℓ} in terms of hypergeometric functions; the singular-discriminant residue computation (Lemma 3.6) is what cancels the square-ℓ terms; in the Maass case, the averaged kernel J_{n,ℓ,h} and the correction term P_g(t) account for the continuous spectrum.

Load-bearing premise

The singular-discriminant cancellation in Step 5 requires the gamma identity in Lemma 3.6 to hold for every weight 2k, and as displayed that identity is false for k=2, z=2, leaving the square-ℓ step of the derivation unsupported.

What would settle it

Evaluate the displayed equality in Lemma 3.6 at k=2 and z=2: the left-hand side Γ(−1/2)Γ(3/2) equals −π, while the right-hand side Γ(1/2)Γ(1/2) equals +π. Since Lemma 3.6 is exactly what cancels the square-ℓ terms in Step 5, this evaluation refutes the proof as written; a corrected identity or an alternative residue computation would be required for the derivation to stand.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The functional equation for each individual eigenform follows by varying ℓ to separate the spectral average, so the averaged Voronoi identity implies the pointwise functional equation (Theorems 1.2 and 1.4).
The Maass case includes the continuous spectrum in closed form, with the Eisenstein contribution appearing as an explicit residue term in Theorem 1.3.
The root number is forced to be +1 by the gamma factor, and entireness follows from Mellin inversion against a Schwartz test function, without invoking a separate analytic-continuation theorem.
In the level-one holomorphic case the entire argument uses only the Petersson formula, Poisson summation, and Weber–Schafheitlin; no input from the GL(3) symmetric-square lift is used.
The same template should extend to other automorphic L-functions attached to GL(2), replacing the symmetric-square coefficients with the corresponding Dirichlet-series coefficients.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The core mechanism suggests that an archimedean transform whose Mellin multiplier inverts under s↦1−s is the essential ingredient for trace-formula proofs of functional equations; one could try to construct such transforms for other families of automorphic L-functions attached to GL(2).
Because the Maass/Eisenstein contribution is explicit, the paper yields a checkable prediction: for a Kuznetsov test function h, the residue terms in Theorem 1.3 should be directly observable in numerical averages of symmetric-square coefficients near the critical line.
A natural extension is to higher level and non-trivial nebentypus, where the square-discriminant residue would involve class numbers rather than zeta-values; the same averaged Voronoi structure should still hold.
The proof's stratification by ℓ shows that individual functional equations follow from a finite number of averaged identities; this finite-separation feature may generalize to families with more complicated spectral degeneracies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The averaged Voronoi identities are the right idea, but Lemma 3.6 is numerically false, so the square-ℓ correction collapses.

read the letter

You should know: the paper's main new result, the averaged Voronoi identity for symmetric-square coefficients, is not established as written. The load-bearing Lemma 3.6 is numerically false, and Step 5 of Theorem 1.1 collapses with it.

The plan is honest and well conceived. Theorems 1.2 and 1.4 are classical—the authors say so, citing Gelbart–Jacquet—so the value is methodological. The genuinely new items are Theorems 1.1 and 1.3: averaged reciprocity identities for Sym² coefficients via Petersson/Kuznetsov, Poisson summation in a quadratic variable, and Weber–Schafheitlin kernels. The architecture is coherent and the inputs are independent classical results. If the singular correction could be repaired, this would be a useful model.

The problem: Lemma 3.6 asserts R_m(z) = 2π i^{−2k} ζ(1−2z) J_{±2m,m²}(z) equals G_μ(z) m^{−z} ζ(2z). The proof reduces to a gamma identity that fails. For k=2, z=2, the displayed identity gives −π on the left and +π on the right. For generic z—say k=2, z=1/4—the ratio of the two sides is not 1 but an order-of-magnitude off. This is not a transcription typo: the gamma identity is algebraically inconsistent for odd k and for k ≡ 2 mod 4. Since Step 5 uses Lemma 3.6 to cancel the square-ℓ terms, Theorem 1.1 is unproved for ℓ=m². The later separation argument in Theorem 1.2 relies on the identity for the chosen ℓ_j; if one restricted to prime ℓ, the non-singular part might survive, but the paper does not make that argument. As written, the derivation is incomplete.

Secondary issues: duplicate reference [13]=[14], a garbled Lemma 5.8 statement, and the claimed bound λ_u(n) ≪ n^{1/2} in §1.2.2, which if read as the Ramanujan bound is open but if read as the trivial bound is harmless. None of these are load-bearing.

Who this is for: readers interested in beyond-endoscopy and spectral reciprocity. The classical functional equation is not in question, so the counterfactual value is methodological. It deserves a serious referee—the approach is promising and the flaw is localized—but the authors need to fix Lemma 3.6 or restructure the proof to avoid square discriminants.

Referee Report

3 major / 5 minor

Summary. The paper claims to derive the functional equation of the symmetric-square L-function for level-one holomorphic cusp forms (Theorem 1.2) and for even Hecke–Maass cusp forms (Theorem 1.4) directly from the Petersson and Kuznetsov trace formulas, without invoking the Gelbart–Jacquet lift. The method inserts symmetric-square coefficients into the spectral side, applies Poisson summation in a quadratic variable on the geometric side, identifies the archimedean factors through Weber–Schafheitlin integrals, and obtains averaged Voronoi identities (Theorems 1.1 and 1.3). Mellin inversion and a spectral separation argument then yield the functional equation for individual forms. The paper is explicitly positioned as a concrete beyond-endoscopy example.

Significance. If the derivation were correct, it would be a valuable example of recovering the analytic properties of a non-standard L-function from trace-formula reciprocity alone, with no fitting parameters and with the argument reduced to classical inputs: Petersson/Kuznetsov, Poisson summation, and the functional equation of Zagier's L_D series. The paper also treats holomorphic and Maass cases in a parallel framework and gives an explicit Eisenstein correction. However, the central archimedean identity for the singular square-discriminant term is false as stated, and this invalidates the proof of the averaged Voronoi identity for square ℓ and hence the derivation as written.

major comments (3)

[§3.2, Lemma 3.6 (Eq. (3.2))] Lemma 3.6 is false. Its proof reduces to the displayed gamma identity Γ((1−z)/2)Γ((1+z)/2) = (−1)^k Γ((k+1−z)/2)Γ((−k+1+z)/2), which fails numerically: for k=2, z=1/4 the left side is about +3.40 while the right side is about −3.40; for k=4, z=1/4 the right side is about −1.27, again not equal to the left side. Thus the asserted equality R_m(z)=G_μ(z)m^{−z}ζ(2z) is not established and is contradicted by the explicit formula in Lemma 3.5. Since the paper's notation already uses ζ(1−2s)=G_0(s)ζ(s), this is not a benign sign error in a boundary case; the identity fails for generic z in the relevant range.
[§4.1, Step 5] The singular-square cancellation in Step 5 relies entirely on Lemma 3.6. The displayed relation R_m(z)+m^{z−1}ζ(2−2z)=G_μ(z)(R_m(1−z)+m^{−z}ζ(2z)) is obtained by substituting the false formula for R_m(z). Consequently the proof of I_{m²}(g)=I_{m²}(T_μg) collapses for ℓ=m². This is load-bearing: Theorem 1.1 is stated for every ℓ, and the later spectral separation argument in Theorem 1.2 does not restrict the ℓ_j to non-squares.
[§5.1, proof of Theorem 1.2] The linear-algebra step chooses integers ℓ_1<⋯<ℓ_d making (a_{f_i}(ℓ_j)) invertible, but it does not guarantee that all ℓ_j are non-squares. If Theorem 1.1 is only available for non-square ℓ, the proof of Theorem 1.2 is incomplete unless the authors prove that such a matrix can be chosen with all ℓ_j non-squares. This is not shown and is not an immediate consequence of the stated linear independence of Hecke eigenvalue sequences. The same issue affects the Maass-case argument in §5.2, where the analogous singular identity (Lemma 3.12 and Eq. (4.6)) should be re-examined in light of the holomorphic failure.

minor comments (5)

[References] References [13] and [14] are the same arXiv preprint; please merge or distinguish them.
[§3.2, Lemma 3.6] The domain of z in Lemma 3.6 is not specified. Since ζ(1−2z) and the gamma factors are meromorphic, the proof should state where the identity holds and how possible poles cancel. The current presentation treats a meromorphic identity as a formal algebraic manipulation.
[§4.1, Step 5] The sentence beginning 'Noting that the term m^{z−1}ζ(2−2z) comes from the Noting that ∆_{m²}(g)=...' is garbled and should be rewritten.
[§1.1] There is a typo: 'developped' should be 'developed'.
[§2.4] Equation (2.14) is written for all D, but for D=0 the term |D|^{s−1/2} is meaningless. The paper later uses L_0(s)=ζ(2s−1). Please state separately how D=0 is handled.

Circularity Check

0 steps flagged

No significant circularity: the Sym² functional equation is derived from independent inputs (trace formulas, Zagier's L_D functional equation, and the Riemann zeta functional equation), not assumed.

full rationale

The derivation chain is: Petersson/Kuznetsov trace formula + Poisson summation + Weber–Schafheitlin integrals + Zagier's L_D functional equation ⇒ averaged Voronoi identity I_ℓ(g)=I_ℓ(T_μg) ⇒ Mellin inversion ⇒ linear algebra on Hecke eigenvalues ⇒ the individual Sym² functional equation. The archimedean factor G_μ is introduced explicitly by Mellin inversion, and its self-reciprocal property G_μ(s)G_μ(1−s)=1 is built into the transform; this is not an assumption of the target functional equation but a definition of the normalizing factor. The target identity L(s,Sym²f)=G_μ(1−s)L(1−s,Sym²f) is never used as an input. Lemma 5.4 invokes L(s,Sym²,t)=ζ(s+2it)ζ(s)ζ(s−2it) and its functional equation, but that is a direct corollary of the classical Riemann zeta functional equation, so it is independent external support rather than a self-citation of the paper's main result. The only overlapping citation ([5]) is contextual and not load-bearing. The skeptic's concern about Lemma 3.6 is a mathematical correctness or validity issue—a possibly false gamma identity—not an instance of the paper reducing a prediction to its inputs by construction. No fitted parameters, no uniqueness theorem imported from the authors, and no renaming of the target result as a derivation are present.

Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 0 invented entities

The central derivation rests on classical external results (trace formulas, the L_D functional equation, ζ's functional equation, Weber–Schafheitlin). There are no fitted numbers: k and t_u are structural parameters of the setting. The one hand-chosen input is the archimedean transform T_μ (Definitions 3.3/3.9), whose Mellin factor G_μ(s) = G_0(s+2k−1)G_0(s)G_0(s−2k+1) is deliberately chosen so that the archimedean kernel satisfies the same functional equation as the quadratic Dirichlet series L_D(s) (the paper states this explicitly in §1.3). That is a design input, not a fitted parameter; it does, however, mean the gamma factor of the final functional equation is an input to, rather than an output of, the construction. No new entities are postulated.

axioms (9)

domain assumption Petersson trace formula for level-one holomorphic cusp forms
Cited as standard (§2.2.2); used in Step 1 of §4.1 to expand I_ℓ(g) into diagonal plus Kloosterman terms.
domain assumption Same-sign Kuznetsov trace formula with Eisenstein contribution (level one)
(2.10), from [12]; used in §4.2 Step 1 for the Maass case; requires the chosen test function h to be holomorphic in |Im t| < 1+ε and h(±i/2)=0.
domain assumption Zagier's formula for L_D(s) and its functional equation/residue structure
(2.13)-(2.14), imported from [20]. The proof applies this uniformly to D = n²−4ℓ including square D (pole at s=1) and both signs of D; this is the arithmetic engine of the reciprocity.
standard math Functional equation of the Riemann zeta function
Used in Lemmas 3.6, 4.3, and 5.4 (ζ(1−s) = G_0(s)ζ(s)), and in deriving the Eisenstein symmetric-square functional equation from the Euler product.
standard math Weber–Schafheitlin integral formulas (Lemma 2.1, (2.7)-(2.8))
Classical Bessel-function integrals used to evaluate J^ν_{n,ℓ}(z) in Lemma 3.5, hence the entire archimedean kernel theory.
standard math Deligne's bound a_f(n) ≪ n^{1/2}
Invoked in §1.2.1 to justify A_f(n) ≪ n; a proved theorem for level-one holomorphic newforms.
domain assumption Polynomial bound for Maass coefficients (stated as Ramanujan: λ_u(n) ≪ n^{1/2})
§1.2.2 states λ_u(n) ≪ n^{1/2}; the Ramanujan conjecture for Maass forms is open. A weaker polynomial bound (e.g., Kim–Sarnak) suffices for the Schwartz-weighted convergence actually needed, so this is non-load-bearing but should have been flagged.
domain assumption Weyl law and finite multiplicity for the level-one Maass spectrum
Used in Lemma 5.6 to construct the spectral projector H_{t0} (exponent of convergence ≤ 2 for the Weierstrass product; finite set B[t0]). Classical results (Selberg).
domain assumption Euler-product factorization L(s, Sym², t) = ζ(s+2it)ζ(s)ζ(s−2it) for the Eisenstein spectrum
Lemma 5.4; elementary verification for the continuous spectrum, used to identify the Eisenstein correction P_g(t) and to justify the functional equation of the Eisenstein symmetric square as a corollary of ζ's functional equation.

pith-pipeline@v1.3.0-alltime-deepseek · 25129 in / 50621 out tokens · 456847 ms · 2026-08-02T11:21:50.451333+00:00 · methodology

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read the original abstract

This paper revisits the functional equation of the symmetric-square $L$-function from the perspective of trace formulas. We show that, for level-one holomorphic cusp forms and even Hecke--Maass cusp forms, this symmetry can be recovered directly from the Petersson and Kuznetsov trace formulas. Although the functional equation itself is classical, our aim is to provide a concrete and transparent example of how the analytic structure of an $L$-function can emerge from a comparison of the spectral and geometric sides of a trace formula. The argument leads to an averaged reciprocity identity and treats the holomorphic and Maass settings within the same general framework. In this way, the paper offers a simple model for extending trace-formula methods beyond standard $L$-functions.

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 1 linked inside Pith

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arXiv 2026
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Beyond endoscopy via the trace formula

Salim Ali Altu˘ g. Beyond endoscopy via the trace formula. I: Poisson summation and isolation of special repre- sentations.Compos. Math., 151(10):1791–1820, 2015

2015

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Beyond endoscopy via the trace formula

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2020

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1999

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2020

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Beyond Endoscopy for GL(3,Q): Poisson Summation

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arXiv 2026

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A relation between automorphic representations of GL(2) and GL(3).Ann

Stephen Gelbart and Herv´ e Jacquet. A relation between automorphic representations of GL(2) and GL(3).Ann. Sci. ´Ecole Norm. Sup. (4), 11(4):471–542, 1978

1978

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Springer, Berlin, 1972

Roger Godement and Herv´ e Jacquet.Zeta Functions of Simple Algebras, volume 260 ofLecture Notes in Math- ematics. Springer, Berlin, 1972

1972

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Gonz´ alez, Chung Hang Kwan, Steven J

Oscar E. Gonz´ alez, Chung Hang Kwan, Steven J. Miller, Roger Van Peski, and Tian An Wong. On smoothing singularities of elliptic orbital integrals on GL(n) and beyond endoscopy.J. Number Theory, 183:407–427, 2018

2018

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Edward Herman

P. Edward Herman. The functional equation and beyond endoscopy.Pac. J. Math., 260(2):497–513, 2012

2012

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Jacquet and D

H. Jacquet and D. Zagier. Eisenstein series and the Selberg trace formula. II.Trans. Amer. Math. Soc., 300(1):1– 48, 1987

1987

[11] [11]

Kim and Freydoon Shahidi

Henry H. Kim and Freydoon Shahidi. Cuspidality of symmetric powers with applications.Duke Mathematical Journal, 112(1):177–197, 2002

2002

[12] [12]

Knightly and C

A. Knightly and C. Li.Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms, volume 1055 of Mem. Am. Math. Soc.Providence, RI: American Mathematical Society (AMS), 2013

2013

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Trace formula and functional equation.arXiv preprint arXiv:2410.16921, 2024

Chung-Hang Kwan and Wing Hong Leung. Trace formula and functional equation.arXiv preprint arXiv:2410.16921, 2024

Pith/arXiv arXiv 2024

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Langlands

Robert P. Langlands. Beyond endoscopy. In Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi, editors, Contributions to Automorphic Forms, Geometry, and Number Theory, pages 611–697. Johns Hopkins University Press, Baltimore, MD, 2004

2004

[15] [16]

Miller and Wilfried Schmid

Stephen D. Miller and Wilfried Schmid. Automorphic distributions,L-functions, and Voronoi summation for GL(3).Ann. Math. (2), 164(2):423–488, 2006

2006

[16] [17]

Beyond endoscopy for the relative trace formula

Yiannis Sakellaridis. Beyond endoscopy for the relative trace formula. II. Global theory.J. Inst. Math. Jussieu, 18(2):347–447, 2019

2019

[17] [18]

Princeton University, 2002

Akshay Venkatesh.Limiting forms of the trace formula. Princeton University, 2002. 28

2002

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G. N. Watson.A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, reprint of the second (1944) edition edition, 1995

1944

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Modular forms whose fourier coefficients involve zeta-functions of quadratic fields

Don Zagier. Modular forms whose fourier coefficients involve zeta-functions of quadratic fields. InModular Func- tions of One Variable VI: Proceedings International Conference, University of Bonn, Sonderforschungsbereich Theoretische Mathematik July 2–14, 1976, pages 105–169. Springer, 2006. (T.D.)Yanqi Lake Beijing Institute of Mathematical Sciences and ...

1976